(
c
)
Characteristics of C
l
vs. α
Angle of Attack, α
C
l
Slope ≈ 2π if α is in radians
and flow incompressible
α = α
0
Angle of
zero lift
Stall
α
C
l
α = α
0
Angle of
zero lift
Cambered airfoil
Symmetric Airfoil
Angle of Zero Lift Depends on Camber
Model for C
l
at Low α (No Stall)
( )
0
2 α α π − =
l
C
• If we know how an airfoil behaves in low speed,
incompressible flow, we can easily estimate how the lift will be
altered in high speed flight (thin airfoil theory)
• This relation works until the Mach number over the airfoil
exceeds 1 somewhere
Incompressible Flow:
Compressible Flow:
( )
2
,
0
2
1 1
2
∞ ∞
−
= −
−
=
M
C
M
C
ible incompress l
l
α α
π
Drag is caused by
• Skin Friction - The air molecules try to drag the airfoil
with them (due to viscosity)
• Form Drag - The flow separates near the trailing
edge, due to the shape of the body; This causes low
pressures, pulling the object back
• Wave Drag - Shock waves form over the airfoil,
converting energy of the flow into heat, causing drag
• …and one more we need to be in 3 dimensions to
talk about…(Induced Drag)
Particles away
from the
airfoil move
unhindered
Skin Friction and the Boundary Layer
This region of low
speed flow is called
the boundary layer
Particles near the
airfoil stick to the
surface, and try to
slow down the
nearby particles
A tug of war results - airfoil is dragged back with the flow
Laminar Flow
• For laminar flow:
– Streamlines move in an orderly fashion - layer by layer
– The mixing between layers is due to molecular motion, and is very slow
– Drag per unit area is proportional to the slope of the velocity profile at
the wall
– Drag is relatively small
Airfoil Surface
This slope
determines the drag
Turbulent Flow
• For turbulent flow:
– highly unsteady, three-dimensional, and chaotic
– It can still be viewed in a time-averaged manner
• For example, at each point in the flow, we can measure velocities
once every millisecond to collect 1000 samples and average
“Time-Averaged”Turbulent Flow
• Velocity varies rapidly near the wall due to increased mixing
• The slope is higher, so skin friction is higher
• Paradoxically: Because separation properties are different,
form drag could actually be lower and max lift higher
Will it be Laminar or Turbulent?
• Reynolds number
• Surface roughness
• State of flow
(if its already turbulent, it’s not going to become laminar)
• Pressure gradient
• …and other factors…
• On a typical wing, flow is laminar near the leading
edge, but eventually trips to turbulent for (typically)
most of the surface
Drag vs. Mach (Compressibility)
Skin Friction
Wave Drag
(mostly)
C
d
M
Other Popular Airfoil Shapes
• NACA 23012
– 5-digit series
• Laminar flow
– Keep more
laminar flow
• Supercritical
– Delay drag
rise (wave drag)